Intro
Issue
TROLL model currently compute leaf lifespan with Reich’s allometry (Reich et al. 1991). But we have shown that Reich’s allometry is underestimating leaf lifespan for low LMA species. Moreover simulations estimated unrealistically low aboveground biomass for low LMA species. We assumed Reich’s allometry underestimation of leaf lifespan for low LMA species being the source of unrealistically low aboveground biomass inside TROLL simulations. We decided to find a better allometry with Wright et al. (2004) GLOPNET dataset.
We tested different models starting from complet model Mcomp (with logged mean): \[ {LL_s}_j \sim \mathcal{logN}({\beta_0} + {\beta_1}_s*{LMA_s}_j^{{\beta_3}_s} - {\beta_2}_s*{Nmass_s}_j^{{\beta_4}_s},\,\sigma)\,\]
\[s=1,...,S_{=4}~, ~~j=1,...,n_s\] \[{\beta_i}_s \sim \mathcal{N}({\beta_i},\,\sigma_i)\,^I, ~~(\beta_i, \sigma, \sigma_i) \sim \mathcal{\Gamma}(0.001,\,0.001)\,^{2I+1}\] We tested models M1 to M9 detailed in following tabs to find the better trade-off between:
- Complexity (and number of parameters)
- Convergence
- Likelihood
- Prediction quality with Root Mean Square Error of Prediction (RMSEP)
RMSEP was computed by fitting a model without a dataset and evaluating it with the same dataset. RMSEP in results are mean RMSEP for each dataset. Results are shown for each models in eahc model tabs and summarized in Results tab.
Table 2: Models summary.| M | Model |
|---|---|
| M1 | \(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA - {\beta_2}_s*N,\sigma)\,\) |
| M2 | \(LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\) |
| M3 | \(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} - {\beta_2}_s*N^{{\beta_4}_s},\sigma)\,\) |
| M4 | \(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA - N,\sigma)\,\) |
| M5 | \(LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\) |
| M6 | \(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\) |
| M7 | \(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA,\sigma)\,\) |
| M8 | \(LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s},\sigma)\,\) |
| M9 | \(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s},\sigma)\,\) |
LL graph
Figure 1: Leaf mass per area (LMA), leaf nitrogen content (Nmass) and leaflifespan (LL). Leaf mass per area (LMA in \(g.m^{-2}\)), leaf nitrogen content (Nmass, in \(mg.g^-1\)) and leaf lifespan (LL in \(months\)) are taken in GLOPNET dataset from Wright et al. (2004).
M1
Model
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA - {\beta_2}_s*N,\sigma)\,\] Maximum likekihood of 22.5686549 and RMSEP of 8.017
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M2
Model
\[ LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of 9.2464903 and RMSEP of 14.773
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M3
Model
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} - {\beta_2}_s*N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of -3.4231955 and RMSEP of 11.409
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M4
Model
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA - N,\sigma)\,\] Maximum likekihood of 15.0709819 and RMSEP of 22.544
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M5
Model
\[ LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of 1.3426982 and RMSEP of 12.484
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M6
Model
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of -3.0373585 and RMSEP of 11.422
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M7
Model
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA,\sigma)\,\] Maximum likekihood of 7.7025968 and RMSEP of 21.384
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M8
Model
\[ LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s},\sigma)\,\] Maximum likekihood of 4.4031261 and RMSEP of 30.281
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
M9
Model
\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s},\sigma)\,\] Maximum likekihood of -7.3836975 and RMSEP of 12.022
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Convergence
Parameters markov chains. Light grey area represents warmup iterations.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
Results
Column
Model M1 seemed to have shown the best trade-off between complexity (\(K=6\) parameters), convergence (see tab M1), likelihood, and prediction quality (see table 2). Figure 2 presents model prediction confidence interval and figure 3 compare Reich’s allometry and model M1 predictions with species functional traits used in TROLL. Nevertheless we will test other predictors gathered from TRY database following a model with the same response curve.
Table 2: Models likelihood and prediction quality.Root mean square errof of prediction (RMSEP) was computed by fitting a model without a dataset and evaluating it with the same dataset. RMSEP in results are mean RMSEP for each dataset.
| ML | RMSEP | |
|---|---|---|
| M1 | 22.569 | 8.016613 |
| M2 | 9.246 | 14.772536 |
| M3 | -3.423 | 11.408979 |
| M4 | 15.071 | 22.544376 |
| M5 | 1.343 | 12.483578 |
| M6 | -3.037 | 11.421916 |
| M7 | 7.703 | 21.383978 |
| M8 | 4.403 | 30.280615 |
| M9 | -7.384 | 12.021887 |
\[LL = 13.062*e^{0.001 *LMA -0.503*Nmass }\]
Column
Figure 2: Leaflifespan predictions for model M1 with leaf mass per area (LMA), and leaf nitrogen content (Nmass). Leaf lifespan (LL in \(months\)) is predicted with model M1 fit. Leaf mass per area (LMA in \(g.m^{-2}\)) and leaf nitrogen content (Nmass, in %) are taken in GLOPNET dataset from Wright et al. (2004). Warning Nmass (resp. LMA) is not constant and depend on the closest point value for left (resp. right) graph.
Figure 3: Leaflifespan predictions for model M1 and Reich’s allometry with leaf mass per area (LMA), and leaf nitrogen content (Nmass). Leaf lifespan (LL in \(months\)) is predicted with model M1 fit or Reich’s allometry (Reich et al. 1991). Leaf mass per area (LMA in \(g.m^{-2}\)) and leaf nitrogen content (Nmass, in %) are taken in BRIDGE dataset used by TROLL (Maréchaux & Chave).
References
Maréchaux, I. & Chave, J. Joint simulation of carbon and tree diversity in an Amazonian forest with an individual-based forest model. Inprep, 1–13.
Reich, P.B., Uhl, C., Walters, M.B. & Ellsworth, D.S. (1991). Leaf lifespan as a determinant of leaf structure and function among 23 amazonian tree species. Oecologia, 86, 16–24.
Wright, I.J., Reich, P.B., Westoby, M., Ackerly, D.D., Baruch, Z., Bongers, F., Cavender-Bares, J., Chapin, T., Cornelissen, J.H.C., Diemer, M. & Others. (2004). The worldwide leaf economics spectrum. Nature, 428, 821–827.